Coalition Graphs

A coalition in a graph $G = (V, E)$ consists of two disjoint sets $V_1$ and $V_2$ of vertices, such that neither $V_1$ nor $V_2$ is a dominating set, but the union $V_1 cup V_2$ is a dominating set of $G$. A coalition partition in a graph $G$ of order $n = |V|$ is a vertex partition $pi = {V_1, V_2, ldots, V_k}$ such that every set $V_i$ either is a dominating set consisting of a single vertex of degree $n-1$, or is not a dominating set but forms a coalition with another set $V_j$. Associated with every coalition partition $pi$ of a graph $G$ is a graph called the coalition graph of $G$ with respect to $pi$, denoted $CG(G,pi)$, the vertices of which correspond one-to-one with the sets $V_1, V_2, ldots, V_k$ of $pi$ and two vertices are adjacent in $CG(G,pi)$ if and only if their corresponding sets in $pi$ form a coalition. In this paper, we initiate the study of coalition graphs and we show that every graph is a coalition graph.